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Properties

Label 91.10.f.b
Level $91$
Weight $10$
Character orbit 91.f
Analytic conductor $46.868$
Analytic rank $0$
Dimension $64$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [91,10,Mod(22,91)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(91, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("91.22");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 91.f (of order \(3\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(46.8682610909\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(32\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 16 q^{2} - 4 q^{3} - 8448 q^{4} - 4016 q^{5} + 1497 q^{6} + 76832 q^{7} - 33654 q^{8} - 196012 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 16 q^{2} - 4 q^{3} - 8448 q^{4} - 4016 q^{5} + 1497 q^{6} + 76832 q^{7} - 33654 q^{8} - 196012 q^{9} - 124744 q^{10} + 104254 q^{11} - 500096 q^{12} + 23624 q^{13} + 76832 q^{14} - 51288 q^{15} - 2070092 q^{16} - 135194 q^{17} + 1380920 q^{18} - 391146 q^{19} + 1682147 q^{20} - 19208 q^{21} - 2771277 q^{22} - 2072050 q^{23} + 8032239 q^{24} + 23897068 q^{25} + 1966761 q^{26} + 2051600 q^{27} + 20283648 q^{28} - 8623630 q^{29} + 2414900 q^{30} + 409456 q^{31} + 30869460 q^{32} + 9155168 q^{33} + 2023812 q^{34} - 4821208 q^{35} - 64248741 q^{36} + 19968248 q^{37} - 132985570 q^{38} + 81838760 q^{39} + 131432816 q^{40} + 43731568 q^{41} - 3594297 q^{42} - 34475548 q^{43} - 207522072 q^{44} + 5649098 q^{45} - 23482400 q^{46} - 80548628 q^{47} + 169512607 q^{48} - 184473632 q^{49} + 7057307 q^{50} - 125344872 q^{51} - 238646397 q^{52} + 70058680 q^{53} + 166462097 q^{54} - 85253262 q^{55} - 40401627 q^{56} - 641553308 q^{57} + 477723014 q^{58} - 34160698 q^{59} + 726578830 q^{60} + 39779802 q^{61} + 280850621 q^{62} + 470624812 q^{63} + 1525670358 q^{64} - 1177975838 q^{65} - 936746468 q^{66} + 673370906 q^{67} - 189564840 q^{68} + 957684708 q^{69} - 599020688 q^{70} + 323250872 q^{71} - 1122296474 q^{72} + 622745480 q^{73} + 26561737 q^{74} - 271694892 q^{75} - 180832666 q^{76} + 500627708 q^{77} - 1459033865 q^{78} + 2992970484 q^{79} + 1591189232 q^{80} - 1063809612 q^{81} + 1871898301 q^{82} - 5399683476 q^{83} - 600365248 q^{84} - 1297906086 q^{85} - 2177158812 q^{86} + 37548244 q^{87} - 4433608251 q^{88} + 2420895886 q^{89} + 11440806726 q^{90} + 457702630 q^{91} + 4676317176 q^{92} - 138248466 q^{93} + 144980485 q^{94} + 233821416 q^{95} - 4734641636 q^{96} + 2936964564 q^{97} + 92236816 q^{98} - 7847662320 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1 −20.8734 + 36.1538i 10.6956 − 18.5253i −615.397 − 1065.90i 1288.61 446.507 + 773.373i 1200.50 + 2079.33i 30007.4 9612.71 + 16649.7i −26897.7 + 46588.2i
22.2 −20.7659 + 35.9676i 92.2265 − 159.741i −606.448 − 1050.40i −1802.37 3830.34 + 6634.34i 1200.50 + 2079.33i 29109.5 −7169.94 − 12418.7i 37427.9 − 64827.0i
22.3 −19.7356 + 34.1831i −73.0287 + 126.489i −522.989 − 905.844i 487.593 −2882.53 − 4992.69i 1200.50 + 2079.33i 21076.8 −824.878 − 1428.73i −9622.94 + 16667.4i
22.4 −19.3238 + 33.4698i 132.534 − 229.556i −490.818 − 850.121i 1551.42 5122.13 + 8871.78i 1200.50 + 2079.33i 18150.3 −25289.1 − 43802.1i −29979.4 + 51925.8i
22.5 −16.9313 + 29.3259i −117.012 + 202.670i −317.339 − 549.648i −1830.35 −3962.33 − 6862.95i 1200.50 + 2079.33i 4154.21 −17542.0 − 30383.6i 30990.3 − 53676.8i
22.6 −16.5748 + 28.7083i 27.4103 − 47.4760i −293.445 − 508.262i −1135.07 908.638 + 1573.81i 1200.50 + 2079.33i 2482.57 8338.85 + 14443.3i 18813.5 − 32586.0i
22.7 −14.3900 + 24.9242i −109.728 + 190.055i −158.145 − 273.915i 2772.83 −3157.99 − 5469.80i 1200.50 + 2079.33i −5632.54 −14239.2 − 24663.0i −39901.0 + 69110.6i
22.8 −13.6001 + 23.5560i 47.4211 − 82.1358i −113.925 − 197.323i 857.041 1289.86 + 2234.11i 1200.50 + 2079.33i −7728.94 5343.97 + 9256.03i −11655.8 + 20188.5i
22.9 −11.0790 + 19.1893i 26.8430 − 46.4934i 10.5132 + 18.2094i −2094.65 594.785 + 1030.20i 1200.50 + 2079.33i −11810.8 8400.41 + 14549.9i 23206.6 − 40194.9i
22.10 −9.39180 + 16.2671i −54.8634 + 95.0263i 79.5881 + 137.851i 964.624 −1030.53 − 1784.94i 1200.50 + 2079.33i −12607.1 3821.51 + 6619.04i −9059.55 + 15691.6i
22.11 −9.21661 + 15.9636i 109.075 − 188.923i 86.1083 + 149.144i 893.937 2010.60 + 3482.45i 1200.50 + 2079.33i −12612.3 −13953.0 − 24167.4i −8239.06 + 14270.5i
22.12 −9.16154 + 15.8683i −89.2168 + 154.528i 88.1322 + 152.649i −200.693 −1634.73 − 2831.43i 1200.50 + 2079.33i −12611.1 −6077.79 − 10527.0i 1838.66 − 3184.65i
22.13 −6.89938 + 11.9501i −21.0558 + 36.4696i 160.797 + 278.509i −1979.45 −290.543 − 503.236i 1200.50 + 2079.33i −11502.6 8954.81 + 15510.2i 13657.0 − 23654.6i
22.14 −3.28755 + 5.69421i −5.26152 + 9.11322i 234.384 + 405.965i 2064.40 −34.5951 − 59.9204i 1200.50 + 2079.33i −6448.66 9786.13 + 16950.1i −6786.82 + 11755.1i
22.15 −0.822191 + 1.42408i 97.4989 − 168.873i 254.648 + 441.063i −785.258 160.325 + 277.692i 1200.50 + 2079.33i −1679.40 −9170.58 − 15883.9i 645.632 − 1118.27i
22.16 −0.0604153 + 0.104642i −56.7378 + 98.2728i 255.993 + 443.392i −1121.96 −6.85566 − 11.8744i 1200.50 + 2079.33i −123.729 3403.14 + 5894.42i 67.7838 − 117.405i
22.17 0.684215 − 1.18509i 83.4624 − 144.561i 255.064 + 441.783i −342.718 −114.212 − 197.822i 1200.50 + 2079.33i 1398.71 −4090.44 − 7084.86i −234.493 + 406.153i
22.18 3.01998 − 5.23076i −121.344 + 210.174i 237.759 + 411.811i 1226.23 732.912 + 1269.44i 1200.50 + 2079.33i 5964.57 −19607.2 − 33960.7i 3703.18 − 6414.10i
22.19 4.34020 − 7.51745i 6.05202 − 10.4824i 218.325 + 378.150i 1591.86 −52.5340 − 90.9916i 1200.50 + 2079.33i 8234.67 9768.25 + 16919.1i 6908.99 − 11966.7i
22.20 5.53284 − 9.58316i −125.872 + 218.017i 194.775 + 337.361i −2058.97 1392.86 + 2412.51i 1200.50 + 2079.33i 9976.27 −21846.2 − 37838.8i −11391.9 + 19731.4i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 22.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.10.f.b ✓ 64
13.c even 3 1 inner 91.10.f.b ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.10.f.b ✓ 64 1.a even 1 1 trivial
91.10.f.b ✓ 64 13.c even 3 1 inner